Question

A test tube weighing $17gf$ floats in alcohol to the level P. When the test tube is made to float in water to the level P, $3gf$ of lead shots are added to it. Find the relative density of the alcohol.

Answer 1

To solve this question, we need to use the Archimedes principle in both the conditions and then take the ratio of the two relations obtained. Then we can calculate the relative density of the liquid from the ratio thus obtained.

Formula used: The formula used in this solution is given as,
${F_B} = \rho gV$
Here, ${F_B}$ is the buoyant force acting on the test tube, $\rho$ is the density of the liquid, $g$ is the acceleration due to gravity, and $V$ is the volume of the test tube.

Complete step by step solution:
Here, we should first calculate the buoyant force and then apply Archimedes Principle according to which the Buoyant Force acting on a body is equal to the weight of the body in equilibrium.
Now, we are given that,
Weight of the test tube $= 17gf$
Let the density of alcohol be ${\rho _a}$
Then, we can write Buoyant Force as,
${F_B} = {\rho _a}gV$
Now applying the Archimedes Principle, we get,
Since, buoyant force is equal to the weight of the body, so,
${\rho _a}gV = 17gf$
Now for the case of water, to get the test tube to float at the same point P as before, we added lead shots weighing 3gf. So, total weight of the test tube becomes,
$3gf + 17gf = 20gf$
So, for water, by applying the Archimedes Principle, we get,
${\rho _w}gV = 20gf$
Here, ${\rho _w}$ is the density of water.
Now taking ratio of both the expressions, we get,
$\dfrac{{{\rho _a}gV}}{{{\rho _w}gV}} = \dfrac{{17gf}}{{20gf}}$
Thus, we get the relative density as,
$R.D. = \dfrac{{{\rho _a}}}{{{\rho _w}}} \\\ \Rightarrow R.D. = \dfrac{{17}}{{20}} \\\$
Hence, we get,
$R.D. = 0.85$
$\therefore$ The relative density of Alcohol is $0.85$ .

Note:
The unit used in this question for weights is the unit of force. Do not get confused or worry about converting it into the SI unit. This is because here in this question, the ratio of the weights is required to be obtained. Due to this, the choice of the unit system becomes unimportant.